Reduced-order Galerkin models of plane Couette flow

TL;DR

This paper develops reduced-order Galerkin models for plane Couette flow using leading controllability modes, achieving a balance between simplicity and accuracy in capturing flow dynamics at moderate Reynolds numbers.

Contribution

It introduces a new approach to model plane Couette flow with low-dimensional systems based on controllability modes, enabling stable long-term simulations.

Findings

Models match DNS statistics at Re=500 and 1200

Number of degrees of freedom ranges from 144 to 600

Models remain numerically stable over large times

Abstract

Reduced-order models were derived for plane Couette flow using Galerkin projection, with orthonormal basis functions taken as the leading controllability modes of the linearised Navier-Stokes system for a few low wavenumbers. Resulting Galerkin systems comprise ordinary differential equations, with a number of degrees of freedom ranging from 144 to 600, which may be integrated to large times without sign of numerical instability. The reduced-order models so obtained are also found to match statistics of direct numerical simulations at Reynolds number 500 and 1200 with reasonable accuracy, despite a truncation of orders of magnitude in the degrees of freedom of the system. The present models offer thus an interesting compromise between simplicity and accuracy in a canonical wall-bounded flow, with relatively few modes representing coherent structures in the flow and their dominant dynamics.